In logic, a manyvalued logic (also multi or multiplevalued logic) is a propositional calculus in which there are more than two truth values. Traditionally, in Aristotle's logical calculus, there were only two possible values (i.e., "true" and "false") for any proposition. An obvious extension to classical twovalued logic is an nvalued logic for n greater than 2. Those most popular in the literature are threevalued (e.g., ukasiewicz's and Kleene's, which accept the values "true", "false", and "unknown"), the finitevalued with more than three values, and the infinitevalued, such as fuzzy logic and probability logic.
History
The first known classical logician who didn't fully accept the law of excluded middle was Aristotle (who, ironically, is also generally considered to be the first classical logician and the "father of logic"^{[1]}). Aristotle admitted that his laws did not all apply to future events (De Interpretatione, ch. IX), but he didn't create a system of multivalued logic to explain this isolated remark. Until the coming of the 20th century, later logicians followed Aristotelian logic, which includes or assumes the law of the excluded middle.
The 20th century brought back the idea of multivalued logic. The Polish logician and philosopher, Jan ukasiewicz, began to create systems of manyvalued logic in 1920, using a third value, "possible", to deal with Aristotle's paradox of the sea battle. Meanwhile, the American mathematician, Emil L. Post (1921), also introduced the formulation of additional truth degrees with n 2, where n are the truth values. Later, Jan ukasiewicz and Alfred Tarski together formulated a logic on n truth values where n 2. In 1932 Hans Reichenbach formulated a logic of many truth values where n infinity. Kurt G del in 1932 showed that intuitionistic logic is not a finitelymany valued logic, and defined a system of G del logics intermediate between classical and intuitionistic logic; such logics are known as intermediate logics.
Examples
Kleene (K_{3}) and Priest logic (P_{3})
Kleene's "(strong) logic of indeterminacy" K_{3} and Priest's "logic of paradox" add a third "undefined" or "indeterminate" truth value I. The truth functions for negation ( ), conjunction ( ), disjunction ( ), implication ( _{K}), and biconditional ( _{K}) are given by:^{[2]}


T 
I 
F 
T 
T 
I 
F 
I 
I 
I 
F 
F 
F 
F 
F 


T 
I 
F 
T 
T 
T 
T 
I 
T 
I 
I 
F 
T 
I 
F 

_{K} 
T 
I 
F 
T 
T 
I 
F 
I 
T 
I 
I 
F 
T 
T 
T 

_{K} 
T 
I 
F 
T 
T 
I 
F 
I 
I 
I 
I 
F 
F 
I 
T 

The difference between the two logics lies in how tautologies are defined. In K_{3} only T is a designated truth value, while in P_{3} both T and I are. In Kleene's logic I can be interpreted as being "underdetermined", being neither true not false, while in Priest's logic I can be interpreted as being "overdetermined", being both true and false. K_{3} does not have any tautologies, while P_{3} has the same tautologies as classical twovalued logic.
K_{3} has additional connectives for conjunction ( _{+}), disjunction ( _{+}) and implication ( _{+}):^{[3]}
_{+} 
T 
I 
F 
T 
T 
I 
F 
I 
I 
I 
I 
F 
F 
I 
F 

_{+} 
T 
I 
F 
T 
T 
I 
F 
I 
I 
I 
I 
F 
T 
I^{[4]}

T 

Belnap logic (B_{4})
Belnap's logic B_{4} combines K_{3} and P_{3}. The overdetermined truth value is here denoted as B and the underdetermined truth value as N.

f _{} 
T 
B 
N 
F 
T 
T 
B 
N 
F 
B 
B 
B 
F 
F 
N 
N 
F 
N 
F 
F 
F 
F 
F 
F 

f_{∨} 
T 
B 
N 
F 
T 
T 
T 
T 
T 
B 
T 
B 
T 
B 
N 
T 
T 
N 
N 
F 
T 
B 
N 
F 

Semantics
Matrix semantics (logical matrices)
Proof theory
Relation to classical logic
Logics are usually systems intended to codify rules for preserving some semantic property of propositions across transformations. In classical logic, this property is "truth." In a valid argument, the truth of the derived proposition is guaranteed if the premises are jointly true, because the application of valid steps preserves the property. However, that property doesn't have to be that of "truth"; instead, it can be some other concept.
Multivalued logics are intended to preserve the property of designationhood (or being designated). Since there are more than two truth values, rules of inference may be intended to preserve more than just whichever corresponds (in the relevant sense) to truth. For example, in a threevalued logic, sometimes the two greatest truthvalues (when they are represented as e.g. positive integers) are designated and the rules of inference preserve these values. Precisely, a valid argument will be such that the value of the premises taken jointly will always be less than or equal to the conclusion.
For example, the preserved property could be justification, the foundational concept of intuitionistic logic. Thus, a proposition is not true or false; instead, it is justified or flawed. A key difference between justification and truth, in this case, is that the law of excluded middle doesn't hold: a proposition that is not flawed is not necessarily justified; instead, it's only not proven that it's flawed. The key difference is the determinacy of the preserved property: One may prove that P is justified, that P is flawed, or be unable to prove either. A valid argument preserves justification across transformations, so a proposition derived from justified propositions is still justified. However, there are proofs in classical logic that depend upon the law of excluded middle; since that law is not usable under this scheme, there are propositions that cannot be proven that way.
Suszko's thesis
Relation to fuzzy logic
Multivalued logic is closely related to fuzzy set theory and fuzzy logic. The notion of fuzzy subset was introduced by Lotfi Zadeh as a formalization of vagueness; i.e., the phenomenon that a predicate may apply to an object not absolutely, but to a certain degree, and that there may be borderline cases. Indeed, as in multivalued logic, fuzzy logic admits truth values different from "true" and "false". As an example, usually the set of possible truth values is the whole interval [0,1]. Nevertheless, the main difference between fuzzy logic and multivalued logic is in the aims. In fact, in spite of its philosophical interest (it can be used to deal with the Sorites paradox), fuzzy logic is devoted mainly to the applications. More precisely, there are two approaches to fuzzy logic. The first one is very closely linked with multivalued logic tradition (Hajek school). So a set of designed values is fixed and this enables us to define an entailment relation. The deduction apparatus is defined by a suitable set of logical axioms and suitable inference rules. Another approach (Goguen, Pavelka and others) is devoted to defining a deduction apparatus in which approximate reasonings are admitted. Such an apparatus is defined by a suitable fuzzy subset of logical axioms and by a suitable set of fuzzy inference rules. In the first case the logical consequence operator gives the set of logical consequence of a given set of axioms. In the latter the logical consequence operator gives the fuzzy subset of logical consequence of a given fuzzy subset of hypotheses.
Research venues
An IEEE International Symposium on MultipleValued Logic (ISMVL) has been held annually since 1970. It mostly caters to applications in digital design and verification.^{[5]} The exists the Journal of MultipleValued Logic and Soft Computing.^{[6]}
See also
 Mathematical logic
 Philosophical logic
 Digital logic
Notes
References
Further reading
General

B ziau J.Y. 1997 What is manyvalued logic ? Proceedings of the 27th International Symposium on MultipleValued Logic, IEEE Computer Society, Los Alamitos, pp. 117 121.
 Malinowski, Gregorz, 2001, ManyValued Logics, in Goble, Lou, ed., The Blackwell Guide to Philosophical Logic. Blackwell.

 Cignoli, R. L. O., D'Ottaviano, I, M. L., Mundici, D., 2000. Algebraic Foundations of Manyvalued Reasoning. Kluwer.


S. Gottwald, A Treatise on ManyValued Logics. Studies in Logic and Computation, vol. 9, Research Studies Press: Baldock, Hertfordshire, England, 2001.



H jek P., 1998, Metamathematics of fuzzy logic. Kluwer. (Fuzzy logic understood as manyvalued logic sui generis.)
Specific

Alexandre Zinoviev, Philosophical Problems of ManyValued Logic, D. Reidel Publishing Company, 169p., 1963.
 Prior A. 1957, Time and Modality. Oxford University Press, based on his 1956 John Locke lectures

Goguen J.A. 1968/69, The logic of inexact concepts, Synthese, 19, 325 373.
 Chang C.C. and Keisler H. J. 1966. Continuous Model Theory, Princeton, Princeton University Press.
 Gerla G. 2001, Fuzzy logic: Mathematical Tools for Approximate Reasoning, Kluwer Academic Publishers, Dordrecht.
 Pavelka J. 1979, On fuzzy logic I: Manyvalued rules of inference, Zeitschr. f. math. Logik und Grundlagen d. Math., 25, 45 52.
 Covers proof theory of manyvalued logics as well, in the tradition of H jek.



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